Measurement 1 of 4 · the referee

Accuracy

Every app was required to ship its own judge: a live mean-squared-error readout comparing the traced wave to an ideal square wave. This is the number that would have exposed faked math. Nobody faked it.

Mean-squared error vs an ideal square wave, read from each app's own readout. Lower is better. Axis runs 0 → 0.05. *Gemini's build had no numeric capture in the recording; its trace visually matched the other three.

What they all implemented

Four separate codebases independently arrived at the same partial sum:

Odd harmonics only, amplitudes falling off as 1/n — the textbook Fourier series of a square wave. Truncating at N terms leaves a floor of irreducible error (plus Gibbs ringing at the jumps), which is why all four correct implementations land at the same ≈0.040 rather than at zero. Converging on the same floor is exactly what four right answers look like.

Why this is the load-bearing measurement

A spread of 0.0402 to 0.04086 is a rounding error between four different authors. That's not luck — it's four models all correctly understanding the same underlying math and executing it correctly. The differences that remain are taste, not correctness.

This is the evidence behind the study's one claim: when quality converges this hard, "which model" stops being the interesting question. See Cost for the variable that survived.

Verify it yourself

Each MSE readout is live — open any build, drag the term slider, and watch the error fall as N rises. The measurement reproduces on your machine, not just in this table.